Sets, Relations & Functions
General
Grade 11
Question:
Let f : R → R be such that for all <span class="math-inline">x \in R</span> (<span class="math-inline">2^{1+x} + 2^{1-x}</span>), f(x) and (<span class="math-inline">3x + 3-x</span>) are in A.P., then the minimum value of f(x) is
<span class="math-inline">0</span>
<span class="math-inline">3</span>
<span class="math-inline">2</span>
<span class="math-inline">4</span>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f⁻¹(x)=(3x-2)/(x-1); g⁻¹(x)=(x+3)/2. Sum=(3x-2)/(x-1)+(x+3)/2=13/2. Solving: multiply out, get quadratic with sum of roots=5.</p><div class="key-concept"><strong>Key Concept:</strong> Explicit inverse + Vieta's sum of roots</div></div>
Correct Answer: 1